direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×Q8○D8, C8.5C24, C4.10C25, D4.7C24, Q8.7C24, D8.10C23, Q16.9C23, SD16.1C23, M4(2).19C23, 2- 1+4⋊8C22, Q8○(C2×D8), D8○(C2×Q8), Q16○(C2×D4), D4○(C2×Q16), C4○D4.40D4, D4.63(C2×D4), Q8.65(C2×D4), (C2×D4).359D4, C4○D8⋊11C22, C8○D4⋊15C22, (C2×Q8).278D4, C2.45(D4×C23), (C2×C8).575C23, (C2×C4).616C24, (C22×Q16)⋊23C2, (C2×Q16)⋊61C22, C4○D4.16C23, C4.127(C22×D4), C23.487(C2×D4), (C2×D8).178C22, (C2×D4).492C23, C8.C22⋊14C22, (C2×Q8).323C23, C22.19(C22×D4), (C22×C8).298C22, (C2×2- 1+4)⋊12C2, (C22×C4).1227C23, (C2×SD16).128C22, (C22×Q8).374C22, (C2×M4(2)).292C22, (C2×Q8)○(C2×D8), (C2×C8○D4)⋊12C2, (C2×C4○D8)⋊32C2, (C2×C4).1115(C2×D4), (C2×C8.C22)⋊35C2, (C2×C4○D4).248C22, SmallGroup(128,2315)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×Q8○D8
G = < a,b,c,d,e | a2=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >
Subgroups: 996 in 704 conjugacy classes, 428 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C22×C8, C2×M4(2), C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, C8.C22, C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2- 1+4, 2- 1+4, C2×C8○D4, C22×Q16, C2×C4○D8, C2×C8.C22, Q8○D8, C2×2- 1+4, C2×Q8○D8
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, C25, Q8○D8, D4×C23, C2×Q8○D8
(1 60)(2 61)(3 62)(4 63)(5 64)(6 57)(7 58)(8 59)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 49)
(1 28 5 32)(2 29 6 25)(3 30 7 26)(4 31 8 27)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)(33 48 37 44)(34 41 38 45)(35 42 39 46)(36 43 40 47)(49 60 53 64)(50 61 54 57)(51 62 55 58)(52 63 56 59)
(1 43 5 47)(2 44 6 48)(3 45 7 41)(4 46 8 42)(9 49 13 53)(10 50 14 54)(11 51 15 55)(12 52 16 56)(17 64 21 60)(18 57 22 61)(19 58 23 62)(20 59 24 63)(25 33 29 37)(26 34 30 38)(27 35 31 39)(28 36 32 40)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 24)(18 23)(19 22)(20 21)(25 30)(26 29)(27 28)(31 32)(33 38)(34 37)(35 36)(39 40)(41 44)(42 43)(45 48)(46 47)(49 56)(50 55)(51 54)(52 53)(57 62)(58 61)(59 60)(63 64)
G:=sub<Sym(64)| (1,60)(2,61)(3,62)(4,63)(5,64)(6,57)(7,58)(8,59)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49), (1,28,5,32)(2,29,6,25)(3,30,7,26)(4,31,8,27)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)(33,48,37,44)(34,41,38,45)(35,42,39,46)(36,43,40,47)(49,60,53,64)(50,61,54,57)(51,62,55,58)(52,63,56,59), (1,43,5,47)(2,44,6,48)(3,45,7,41)(4,46,8,42)(9,49,13,53)(10,50,14,54)(11,51,15,55)(12,52,16,56)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(25,33,29,37)(26,34,30,38)(27,35,31,39)(28,36,32,40), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,24)(18,23)(19,22)(20,21)(25,30)(26,29)(27,28)(31,32)(33,38)(34,37)(35,36)(39,40)(41,44)(42,43)(45,48)(46,47)(49,56)(50,55)(51,54)(52,53)(57,62)(58,61)(59,60)(63,64)>;
G:=Group( (1,60)(2,61)(3,62)(4,63)(5,64)(6,57)(7,58)(8,59)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,49), (1,28,5,32)(2,29,6,25)(3,30,7,26)(4,31,8,27)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)(33,48,37,44)(34,41,38,45)(35,42,39,46)(36,43,40,47)(49,60,53,64)(50,61,54,57)(51,62,55,58)(52,63,56,59), (1,43,5,47)(2,44,6,48)(3,45,7,41)(4,46,8,42)(9,49,13,53)(10,50,14,54)(11,51,15,55)(12,52,16,56)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(25,33,29,37)(26,34,30,38)(27,35,31,39)(28,36,32,40), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,24)(18,23)(19,22)(20,21)(25,30)(26,29)(27,28)(31,32)(33,38)(34,37)(35,36)(39,40)(41,44)(42,43)(45,48)(46,47)(49,56)(50,55)(51,54)(52,53)(57,62)(58,61)(59,60)(63,64) );
G=PermutationGroup([[(1,60),(2,61),(3,62),(4,63),(5,64),(6,57),(7,58),(8,59),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,41),(24,42),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,49)], [(1,28,5,32),(2,29,6,25),(3,30,7,26),(4,31,8,27),(9,17,13,21),(10,18,14,22),(11,19,15,23),(12,20,16,24),(33,48,37,44),(34,41,38,45),(35,42,39,46),(36,43,40,47),(49,60,53,64),(50,61,54,57),(51,62,55,58),(52,63,56,59)], [(1,43,5,47),(2,44,6,48),(3,45,7,41),(4,46,8,42),(9,49,13,53),(10,50,14,54),(11,51,15,55),(12,52,16,56),(17,64,21,60),(18,57,22,61),(19,58,23,62),(20,59,24,63),(25,33,29,37),(26,34,30,38),(27,35,31,39),(28,36,32,40)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,24),(18,23),(19,22),(20,21),(25,30),(26,29),(27,28),(31,32),(33,38),(34,37),(35,36),(39,40),(41,44),(42,43),(45,48),(46,47),(49,56),(50,55),(51,54),(52,53),(57,62),(58,61),(59,60),(63,64)]])
44 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4T | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
44 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
type | + | + | + | + | + | + | + | + | + | + | - |
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | Q8○D8 |
kernel | C2×Q8○D8 | C2×C8○D4 | C22×Q16 | C2×C4○D8 | C2×C8.C22 | Q8○D8 | C2×2- 1+4 | C2×D4 | C2×Q8 | C4○D4 | C2 |
# reps | 1 | 1 | 3 | 3 | 6 | 16 | 2 | 3 | 1 | 4 | 4 |
Matrix representation of C2×Q8○D8 ►in GL6(𝔽17)
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 13 | 0 | 0 | 0 |
0 | 0 | 0 | 13 | 0 | 0 |
0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 0 | 4 |
16 | 13 | 0 | 0 | 0 | 0 |
9 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 3 | 14 | 0 | 0 |
0 | 0 | 3 | 3 | 0 | 0 |
0 | 0 | 0 | 0 | 3 | 14 |
0 | 0 | 0 | 0 | 3 | 3 |
16 | 13 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 3 | 3 | 0 | 0 |
0 | 0 | 3 | 14 | 0 | 0 |
0 | 0 | 0 | 0 | 14 | 14 |
0 | 0 | 0 | 0 | 14 | 3 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,9,0,0,0,0,13,1,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[16,0,0,0,0,0,13,1,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,14,3] >;
C2×Q8○D8 in GAP, Magma, Sage, TeX
C_2\times Q_8\circ D_8
% in TeX
G:=Group("C2xQ8oD8");
// GroupNames label
G:=SmallGroup(128,2315);
// by ID
G=gap.SmallGroup(128,2315);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,-2,448,477,456,521,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=e^2=1,c^2=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^3>;
// generators/relations